This is of course true, for any scheme and any point. For any point $y\in X$, you take an affine neighborhood $U$ of $y$, if $U$ contains $y$ then you know $f_*O_{X,x}$ is flat at y, if $U$ does not cantain $y$ then the stalk of $f_*O_{X,x}$ at $y$ is 0, still flat.
The reason that the sheaf is flat is that $Spec(O_{X,x})\to X$ is affine and flat. In general if $f: X\to Y$ is flat and affine then $f_*O_X$ is $O_Y-$flat. This is obvious.
But I don't think it is true that $f: X\to Y$ is flat implies $f_*O_X$ is $O_Y-$flat. I would be happy if someone can back me up with a counter example here.